Elliptical curves and isogenies
- Effective arithmetic of finite bodies and their extensions
– Extension towers, composita, morphisms.
– Algebraic closure of GF (p).
– Geometric applications: torsion, isogenies, Tate module.
- Isogenies, couplings
– Calculation of explicit isogenies: Vélu formulas, Elkies formulas, higher genus.
– Isogeny volcanoes, isogeny graphs.
- Cryptographic applications
– Homomorphic encryption.
– Post-quantum cryptography.
Advanced network reduction algorithms
- Since the invention of LLL, many improvements have been proposed:
– Algorithms which provide reduced databases of the same quality as LLL but faster -> study their implementation on massively parallel computers.
– Algorithms which aim to produce better quality networks than LLL -> obtain bases whose first vector is the shortest vector of the network.
- Implementation of network reduction attacks
– Build an effective and highly configurable toolbox integrating all the standard cryptanalyses using network reduction, including those based on finding small roots using the Coppersmith method.